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This paper considers a distributed stochastic optimization problem where thegoal is to minimize the time average of a cost function subject to a set ofconstraints on the time averages of a related stochastic processes calledpenalties. We assume that a delayed information about an event in the system isavailable as a common information at every user, and the state of the system isevolving in an independent and non-stationary fashion. We show that anapproximate Drift-plus-penalty (DPP) algorithm that we propose achieves a timeaverage cost that is within some positive constant epsilon of the optimal costwith high probability. Further, we provide a condition on the waiting time forthis result to hold. The condition is shown to be a function of the mixingcoefficient, the number of samples (w) used to compute an estimate of thedistribution of the state, and the delay. Unlike the existing work, the methodused in the paper can be adapted to prove high probability results when thestate is evolving in a non-i.i.d and non-stationary fashion. Under mildconditions, we show that the dependency of the error bound on w is exponential,which is a significant improvement compared to the exiting work.

A Probably Approximately Correct Answer to Distributed Stochastic Optimization in a Non-stationary Environment